Parabolics and simple roots for a special unitary group: reference request

Question

I am looking for a reference where the relative root system, the relative system of simple roots, and parabolic $\Bbb R$-subgroups for the real algebraic group ${\rm SU}(p,q)$ are explicitly computed.

More generally, let $F$ be a field of characteristic 0, $L/F$ a quadratic extension, and $H$ be an $L/F$-Hermitian form on $L^n$. Write $G={\rm SU}(L^n,H)$, which is a semisimple $F$-group. I am looking for a reference where the relative system of simple roots and the conjugacy classes of $F$-parabolics in $G$ are explicitly computed. Here "relative" means with respect to a maximal split $F$-torus.

The case of a special orthogonal group is treated in Borel's 1966 paper "Linear algebraic groups" in the Boulder proceedings "Algebraic Groups and Discontinuous Subgroups", Proc. Sympos Pure Math. vol. 9. Borel writes that for an orthogonal group, the $F$-paraboics are the stabilizers of the $F$-rational isotropic flags.

Question. Is it true for $G={\rm SU}(L^n,H)$ that the $F$-parabolics in $G$ are the stabilizers of the $F$-rational isotropic flags in $(L^n, H)$ ?

Answer 1

I have edited in some remarks from the comments (1 2 3). To start, I am used to thinking of the stabiliser of a self-dual flag, but of course every self-dual flag has an isotropic flag as its “bottom half”, and every isotropic flag can be completed to a self-dual flag by tossing the duals “on top”. (Yu - An introduction to explicit Bruhat–Tits theory is where I first encountered the description of Bruhat–Tits buildings of symplectic groups in this ‘self-dual’ language.)

$\DeclareMathOperator\tr{tr}$Write $L^n = V^* \oplus \overline V \oplus W$ for some $L$-vector space $V$, where $(W, H)$ is anisotropic and $H(v^* \oplus \overline v) = \tr_{L/F} \langle v^*, v\rangle$. A choice of $L$-basis $(v_i)_{i \in I}$ for $V$ gives rise to a split torus $S$ in $G$, consisting of the transformations that preserve the $F$-line through each basis vector. Then $S$ is a maximal split torus in $G$, since its centraliser in $G$ is the product of the anisotropic group $\operatorname{SU}(W, H)$ with the torus consisting of the transformations that preserve the $L$-line through each basis vector.

Write $(v_{i^*})_{i \in I}$ for the basis of $V^*$ dual to $(v_i)_{i \in I}$. Put $J = I \cup I^*$.

For each $i \in J$, the map $a_i : S \to \operatorname{GL}_1$ that sends $s \in S$ to $s v_i/v_i$, in the hopefully obvious notation, is a relative root if $W \ne 0$; and its root space consists of all those skew-adjoint endomorphisms of $L^n$ that annihilate all $v_{i'}$ with $i' \ne i, i^*$, and that carry $W$ into $L v_i$.

For each pair $i, j \in J$ such that $i \ne j$, the map $a_{i j} = a_i - a_j$ is a relative root; and its root space is the set of all skew-adjoint endomorphisms of $L^n$ that annihilate all $v_{j'}$ with $j' \ne j, j^*$, and that carry $L v_j$ into $L v_i$. (Note that $a_{i i^*} = 2a_i$, so, to get the full root algebra for $a_i$—if it is a relative root—we need to take the sum of the $a_i$ and $a_{i i^*}$ root spaces.)

Since these spaces, together with $\operatorname C_{\mathfrak{su}(L^n, H)}(S)$, span $\mathfrak{su}(L^n, H)$, we have found all relative roots. The relative root system is of type $\mathsf C_{\dim(V)}$ if $W = 0$ and $\mathsf{BC}_{\dim(V)}$ if $W \ne 0$. If we identify $I$ with $\{1, \dotsc, m\}$, then one choice of simple roots is the union of $\{a_{i(i + 1)} \mathrel: 1 \le i < m\}$ with $\{a_{m m^*}\}$ if $W = 0$, or with $\{a_m\} if $W \ne 0$.

I will now appeal to the parameterisation of (rational) parabolics by (rational) cocharacters, according to which we attach to a (rational) cocharacter $\lambda$ of $G$ the parabolic $P_G(\lambda) = \{g \in G \mathrel: \text{$\lim_{t \to 0} \lambda(t)g\lambda(t)^{-1}$ exists}\}$. (This parameterisation has the advantage that it also singles out a Levi component $M_G(\lambda) = \operatorname{Cent}_G(\lambda)$ of $P_G(\lambda)$.)

Fix a rational cocharacter $\lambda$ of $G$. After replacing it by a $G$-conjugate, we may assume that it takes values in $S$; and, after replacing it by a further Weyl conjugate, we may assume that $\langle a_{i j^*}, \lambda\rangle \ge 0$ for all $i, j \in I$. Then $P_G(\lambda)$ is the stabiliser of the flag $$ \Bigl\{\bigoplus_{\substack{j \in I \\ \langle a_{i j}, \lambda\rangle \ge 0}} L v_j \mathrel: i \in I\Bigr\} $$ (where I've put $a_{ii} = 0$ for each $i \in J$).

Since every isotropic flag can be conjugated into $V$, we have shown that the parabolics are precisely the stabilisers of isotropic flags.

Answer 2

A slightly off-target answer, but possibly of some use. First, indeed, there is a large population who deliberately take the viewpoint, attributed to Harish-Chandra, that everything should be done for arbitrary reductive/semi-simple groups, not just "examples". Thus, as @LSpice's comment, "there's just one reductive group, $G$". :)

But, with some substance for the theory of integral representations of automorphic $L$-functions, the obvious/natural relations of classical groups prove non-trivial theorems... that do not have "intrinsic" analogues, in any useful sense, apparently.

So! "The classical groups". Over $\mathbb C$, there are really just $3$: general or special linear, orthogonal (nevermind the parity of dimension), and symplectic. Over $\mathbb R$ we have the subdivision...

Over $\mathbb R$, we can have Sylvester's Inertia theorems for the real forms that are described by "signatures", the simplest ones being $O(p,q)$, $U(p,q)$, $Sp^*(p,q)$. Yes, modeled by reals, complex, and quaternions. In light of Weil's "Classical groups/algebras with involutions", this is not a coincidence...

So, operationally, there are two types of classical groups: general-linear, and isometry (or similitude). In general-linear, parabolics are stabilizers of flags. In isometry... groups, parabolics are stabilizers of totally isotropic flags.

Choice of Levi component is equivalent to choice of complementary isotropic flag.

One can continue, as desired, to be able to re-specify "roots", etc.

Witt's theorem(s) show that there is only one isomorphism class of choices...

The "physicality" of Inertia Theorems as a part of "classification over $\mathbb R$" is (to my mind) use of inequalities, the intermediate value theorem, and such. Yes, this is all subsumed by the less-"physical" Witt-theorem business, but some of it is (to my mind) much easier to believe.

Yes, for example, Witt's Thm assures us that the leftover, after removing $W\oplus W'$ for maximal totally isotropic $W$ and $W'$ complementing each other, is independent of choices.

The latter bit is perhaps the easiest thing to talk about from a classical-groups viewpoint, but a bit clumsy (so far as I can understand) instrinsically. Anisotropic groups in the Levi component(s) of minimal ($\mathbb R$-) parabolics.

(I must confess that the times I've tried to give courses on "Lie theory" or "algebraic groups" emphasizing "classical groups", I've met considerable resistance from a considerable fraction of the audience, who could not find corroboration for the viewpoint "in the wild".)

After quite a few years of looking at reductive groups from both ends... I do finally think that even the basic facts are fairly well determined by the smallish number of data points we have from the split and quasi-split classical groups. Perhaps more usefully, the phenomena beyond that are easy to exemplify among classical groups, but somewhat clumsy to get-under-the-umbrella of intrinsic description, it seems to me.

(A funny business...)

(To be clearer: A. Weil's "Algebras with involutions and the classical groups"...)